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REVIEW 3 major objections 4 minor 27 references

Data-Enabled Predictive Control for Nonlinear Systems Based on a Koopman Bilinear Realization

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves a nonlinear extension of the classical Fundamental Lemma: for systems with an exact Koopman bilinear realization, persistently exciting input/state data represent every trajectory, enabling direct data-driven predictive…

desk verdict A legitimate formal extension of the fundamental lemma to Koopman bilinear realizations, but the converse is a nonlinear condition, so the control problem stays nonconvex and the numerical claims are stronger than the evidence. read the letter →

arxiv 2505.03346 v1 pith:KY57OM7T submitted 2025-05-06 math.OC

Zuxun Xiong , Zhenyi Yuan , Keyan Miao , Han Wang , Jorge Cortes , Antonis Papachristodoulou This is my paper · ORCID
classification math.OC MSC 93C1093B3093C5537N35
keywords data-enabledpredictivecontrolKoopmanbilinearrealizationfundamentallemmanonlinearcontrol-affinesystemspersistencyofexcitationdata-drivenExtendedDynamicModeDecomposition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that predictive control for nonlinear systems can be designed directly from raw input/state data, without first identifying an approximate model. It does this by extending the classical Fundamental Lemma of behavioral systems theory to nonlinear control-affine systems, using the Koopman bilinear realization: if an exact finite-dimensional lifting exists and the collected data are persistently exciting, any possible trajectory can be written as a linear combination of data Hankel matrices. On that foundation the authors build a data-enabled predictive control formulation that works in the lifted state space and uses slack variables and regularizers to tolerate the approximation error that appears when no exact finite-dimensional Koopman realization exists. A sympathetic reader would care because this removes the EDMD identification step and, in the reported examples, achieves lower total control cost and better robustness than existing Koopman-based methods.

What carries the argument

The central object is the Koopman bilinear realization (KBR), a lifting $z=\Psi(x)$ such that the lifted state evolves as $z_{k+1}=A z_k+B u_k+H(z_k\otimes u_k)$ with $x_k=C z_k$. The argument rides on the block Hankel matrix $G_L(T)=[Z,U,V]$, where $V$ is the Hankel matrix of the bilinear products $z_i\otimes u_i$; full row rank of $G_L(T)$ is the persistency-of-excitation condition in Definition 2. Lemma 2 uses the identity $Z=[O_L,P_L,Q_L]G_L(T)$ to show that every $L$-long trajectory is a data combination, and conversely the constraint $v=Vg$ keeps combinations inside the nonlinear behavior. In the DeePC formulation, the same Hankel matrices become equality constraints that replace the unknown dynamics.

What would settle it

For a system without an exact KBR, such as (8) with basis $\{x_1,x_2,\cos x_1,\sin x_1\}$, collect one long persistently exciting dataset and compute, for a later trajectory $(\bar{x},\bar{u})$, the minimum residual $\min_g \|[X;U]g-[\bar{x};\bar{u}]\|$ subject to $v=Vg$; a residual above numerical noise demonstrates that Lemma 2's premise of exactness is required, so the method's success then rests on regularization rather than the theorem.

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Extended reading notes

Core claim

The paper's central discovery is Lemma 2: under Assumption 1, if input/state data from the nonlinear system are L-persistently exciting in the sense of Definition 2, then every L-long input/state trajectory can be written as the product of the data Hankel matrices $[X;U]$ with some vector $g$, and conversely any $g$ that respects the bilinear-product constraint $v=Vg$ produces a valid trajectory. The condition requires the block Hankel matrix $[Z;U;V]$ built from lifted states, inputs, and products $z\otimes u$ to have full row rank. When the Koopman bilinear term is zero, the constraint on $v$ disappears and the lemma specializes to the exact Koopman-linear case, recovering the classical fundamental lemma as well as earlier nonlinear extensions. The authors then formulate the DeePC problem in the lifted state, with the data representation replacing the explicit dynamics and a slack variable absorbing approximation error, and report numerical evidence that this direct formulation improves optimality over EDMD-based MPC and improves robustness over input-output Koopman DeePC when the realization is inexact.

Load-bearing premise

The result stands on Assumption 1, that the chosen finite set of lifting functions yields an exact Koopman bilinear realization, and since the authors note most nonlinear systems do not admit such a realization, the formal guarantee covers only systems whose chosen observables form an invariant subspace.

Editorial extensions

If this is right

  • For any control-affine system with an exact finite-dimensional KBR, all open-loop input/state behaviors are encoded in one persistently exciting data set, so no EDMD identification or explicit model is required for trajectory generation.
  • A predictive controller can be designed directly in the lifted space, where the equality constraints replace the dynamics and choosing the lifting functions to include the cost and constraints turns the nonlinear MPC problem into an optimization with bilinear constraints.
  • When the system admits an exact Koopman linear realization, the proposed lemma and DeePC reduce to the classical linear case, making the new result a strict generalization of both the classical Fundamental Lemma and the exact-KLR nonlinear DeePC.
  • In systems without an exact KBR, formulating the DeePC in the lifted input/state space is claimed to be more robust than input-output Koopman DeePC, with lower total cost in the reported Van der Pol case.
  • When an exact KBR exists, direct KB-DeePC matches EDMD-based KB-MPC in the numerical study, confirming that the data representation does not sacrifice prediction quality.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because Assumption 1 holds exactly when the chosen observable space is invariant under the Koopman operator, a practical route to widening the method's valid regime is to learn the lifting functions from data alongside the Hankel representation, rather than fixing them in advance.
  • The residual of the constraint $v=Vg$ on fresh data could serve as a measurable certificate of how far a system is from admitting an exact KBR, letting the regularization strength be tuned online.
  • The bilinear DeePC problem is non-convex, so global optimality is not guaranteed; alternating-minimization or branch-and-bound treatments of the constraint $v_i=z_i\otimes u_i$ could restore the kind of certificates available in the linear case.
  • The rank condition in Definition 2 suggests a testable trade-off: as the lifting dimension $n_z$ and input dimension $n_u$ grow, the amount of persistently exciting data needed for full row rank may grow as well, which would determine when the direct method is practical.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper extends Willems' Fundamental Lemma to nonlinear control-affine systems using the Koopman bilinear realization (KBR). Under an exact finite-dimensional KBR assumption and a persistency-of-excitation condition, it claims that any input/state trajectory of the nonlinear system can be represented by data Hankel matrices of lifted states, inputs, and bilinear products (Lemma 2). It then proposes a Data-Enabled Predictive Control (DeePC) formulation with bilinear constraints and regularization to handle inexact KBRs, and presents three numerical studies (Van der Pol oscillator with inexact KLR, a system with an exact KBR, and a system with neither exact KLR nor exact KBR) comparing against EDMD-based Koopman MPC and KLR-based DeePC. The KLR-based case is discussed as a convex special case.

Significance. If the main result were correct as stated, the paper would provide a meaningful step toward direct data-driven predictive control for nonlinear systems, bypassing explicit Koopman model identification. The KBR-based fundamental lemma is a nontrivial extension of the bilinear fundamental lemma of [13] and generalizes the KLR-based result in [18]. The numerical examples, while limited, show plausible performance gains. However, the exact-KBR assumption is restrictive, the converse in Lemma 2 is incomplete as stated, and the nonconvex DeePC problem lacks global optimality guarantees, so the 'improved optimality' and 'advanced robustness' claims are stronger than the evidence supports. With corrections and more careful claims, the contribution could be useful to the data-driven control community.

major comments (3)
  1. [Section III.A, Lemma 2(ii)] The converse statement in Lemma 2(ii) is not valid as stated. Condition (12) with v[1,L] = V_{1,L,T-L+1}g is a linear equation, but for (Z_{1,L+1}g, U_{1,L}g) to be a valid trajectory of the KBR (7), the additional input sequence v[1,L] must equal z_i ⊗ u_i where z_i and u_i are the corresponding entries of Z_{1,L}g and U_{1,L}g. The proof of (ii) only checks the block equation [O_L, P_L, Q_L][z(1); u; v] = Z_{2,L+1}g; it does not verify the quadratic coupling v_i = z_i ⊗ u_i for the recursively defined states. The full-row-rank condition in Definition 2 guarantees existence of a g for part (i) but does not imply that every g satisfying (12) lies in the required variety. The DeePC formulation (16) correctly includes the missing coupling as constraint (16d), so the algorithmic content is preserved, but the lemma as stated overclaims the fundamental-lemma equivalence. Please restrict the converse to g satisfying v_{1,L} = (Z_{1,L}g) ⊗ (U_{1,L}g) and adapt the proof accordingly.
  2. [Section III.B, Eq. (16); Section IV.B] The DeePC problem (16) is nonconvex due to the bilinear constraints (16d), and the paper provides no global optimality guarantee. Even under the exact-KBR Assumption 1, it is not established that a global solution of (16) is equivalent to the original nonlinear MPC (9). The claim of 'improved optimality' in the abstract and in Table II is therefore an empirical observation based on local solutions returned by fmincon, not a theoretical result. The discrepancy between KB-MPC and KB-DeePC in Table II for (N,kc)=(20,20) is direct evidence of the gap. Please state precisely what (16) computes (e.g., a KKT point) and either provide an equivalence analysis under exact KBR or soften the optimality claims to 'local optimality in the lifted data space'.
  3. [Section IV.C; Abstract] The abstract's claim of 'advanced robustness to finite Koopman approximation errors' is supported only by two numerical examples (Sections IV.A and IV.C) without any formal analysis. The paper itself states in Section III.A that a formal treatment of the approximation error e∞_k is left to future work. While empirical demonstration is a legitimate contribution, the robustness claim should be explicitly qualified as heuristic, or the authors should provide an error bound or robustness certificate for the regularized DeePC solution. As written, the claim is stronger than the evidence provided.
minor comments (4)
  1. [Section III.A, proof of Lemma 2] The dimensions of the block matrices P_L and Q_L appear to be misprinted: with B ∈ R^{nz×nu} and H ∈ R^{nz×nz nu}, one should have P_L ∈ R^{nzL×nuL} and Q_L ∈ R^{nzL×nz nu L}, not the dimensions stated in the text.
  2. [Section III.B, Eq. (16)] The objective in (16a) writes z^T Q z + u^T R u and then mentions coefficients q and r, but the linear terms q^T z and r^T u from the original MPC (9) are not present in (16a). Please either include them or remove the reference to q and r.
  3. [Definition 2] The notation v_{[1,T]} is used before it is defined; please explicitly define v_i = z_i ⊗ u_i for i = 1,...,T in the text preceding Definition 2.
  4. [Section IV.A, Figure 2] The caption says 'The black curves represent the open-loop trajectories,' but the other curves are not fully labeled in the caption; please make the figure legend self-contained by identifying which color corresponds to which method.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Lemma 2 is derived from the exact-KBR and persistency-of-excitation assumptions; the cited bilinear-lemma work is reproduced and extended, not invoked as a black box.

full rationale

The paper's central claim, Lemma 2, is a self-contained derivation from its explicit assumptions: Assumption 1 (exact finite-dimensional Koopman bilinear realization), Definition 2 (full-row-rank data matrix G_L(T)=[Z;U;V]), and the bilinear lifted dynamics (7). The proof constructs g from the full-row-rank condition and verifies both directions using the block matrices [O_L,P_L,Q_L], so the representability of any trajectory is an algebraic consequence of the stated PE condition rather than a fitted or pre-supposed result. The DeePC formulation (16) then uses the same representability condition as constraints (16b)-(16d); no parameter is fitted to the trajectories that are later claimed as predictions. The paper does cite its own prior work, notably [13] (Yuan and Cortes) and [18] (Shang, Cortes, and Zheng), but the bilinear fundamental lemma of [13] is re-derived in the proof, and [18] is used only to identify a special case (exact KLR) recovered by the new result. These self-citations are therefore not load-bearing. The acknowledged limitations -- most nonlinear systems do not admit exact finite-dimensional Koopman realizations, and the KBR-based DeePC is nonconvex and solved without global optimality guarantees -- are correctness/scope caveats, not circular reasoning. No specific reduction of a derived claim to its inputs or to an unverified self-citation is present in the paper.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. The central tuning choices are the lifting basis functions and the regularization weights, both selected by hand. The theory rests on the existence of an exact finite-dimensional KBR and on a full-row-rank data condition, with the inexact case handled only by heuristics and regularization.

free parameters (2)
  • Regularizer weight lambda_z = lambda_z = 10 in Van der Pol example
    Appears in (16) as the penalty on the slack variable sigma_z. The value is hand-selected and influences the claimed improvement in Table I.
  • Regularizer weight lambda_g = lambda_g = 0.01 in Van der Pol example
    Penalty on ||g||_2^2 in (16). This is a tunable parameter chosen by the authors and it affects the numerical comparisons.
assumptions (4)
  • domain assumption There exists an exact finite-dimensional Koopman bilinear realization (KBR) of the nonlinear system, with a known lifting function set Psi.
    Assumption 1 in Section III.A is the load-bearing premise of Lemma 2 and of the DeePC derivation. The paper itself states that most nonlinear systems do not admit such an exact finite-dimensional realization, so this assumption restricts the class of systems covered by the theory.
  • domain assumption The collected input/state data satisfy the L-persistently exciting condition in Definition 2, i.e., the matrix GL(T) is full row rank.
    Full row rank of GL(T) is what makes the representation in Lemma 2 possible. The paper does not provide a practical procedure to verify this condition in the numerical examples, nor an explicit guarantee that the chosen data generation achieves it.
  • domain assumption The lifted state z = Psi(x) together with the output map x = Cz exactly reproduces the trajectories of the original nonlinear system when the exact KBR exists.
    This is part of the definition of an exact KBR, invoked in the proof of Lemma 2 and in the formulation of the DeePC constraints. It is a standard Koopman-theory assumption, but it is still an assumption that may fail for arbitrary nonlinear systems.
  • standard math For control-affine systems, infinite-dimensional Koopman bilinear realizations always exist, which motivates the approximate finite-dimensional setting.
    The paper cites [20] for this result and uses it only as motivation for the approximation-error discussion, not as a step in the proof of Lemma 2. The finite-dimensional exact case still requires Assumption 1.

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Pith. "Pith review of Data-Enabled Predictive Control for Nonlinear Systems Based on a Koopman Bilinear Realization." pith.science (2026). https://pith.science/paper/KY57OM7T

@misc{pith2026250503346,
  author       = {Pith},
  title        = {Pith review of: Data-Enabled Predictive Control for Nonlinear Systems Based on a Koopman Bilinear Realization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KY57OM7T}},
  note         = {Machine review of arXiv:2505.03346}
}
read the original abstract

This paper extends the Willems' Fundamental Lemma to nonlinear control-affine systems using the Koopman bilinear realization. This enables us to bypass the Extended Dynamic Mode Decomposition (EDMD)-based system identification step in conventional Koopman-based methods and design controllers for nonlinear systems directly from data. Leveraging this result, we develop a Data-Enabled Predictive Control (DeePC) framework for nonlinear systems with unknown dynamics. A case study demonstrates that our direct data-driven control method achieves improved optimality compared to conventional Koopman-based methods. Furthermore, in examples where an exact Koopman realization with a finite-dimensional lifting function set of the controlled nonlinear system does not exist, our method exhibits advanced robustness to finite Koopman approximation errors compared to existing methods.

Figures

Figures reproduced from arXiv: 2505.03346 by the authors.

Figure 1
Figure 1. Prediction comparison between the KLR and the KBR (with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Closed-loop trajectories starting from x(0) = [−0.8, 0.4] under different methods after 400 steps (left) and 1000 steps (right). slightly higher cost than the Koopman linear MPC, as the Koopman operator enables global linearization of the system dynamics, leading to more accurate prediction and optimiza￾tion of future states and inputs over the prediction horizon N. The proposed KL-DeePC achieves stabilization with … view at source ↗
Figure 4
Figure 4. Comparison of state responses under different control methods. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

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Reviewed August 15, 2026 · model on record in the stance chip above.